ON THE DEGREE OF CONVERGENCE OF THE GRAM ......Gram-Charlier series, which, in addition to having an...

ON THE DEGREE OF CONVERGENCE OF THEGRAM-CHARLIER SERIES*

BY

W. E. MILNE

The degree of convergence of Fourier series and of polynomial approxi-

mations has been extensively studied by Jackson,f and similar investigations

have been made for series of Legendre polynomials,! for Sturm-Liouville

series,§ for Laplace's series,|| for Birkhoff's series,^ and for other related

problems.** In all these cases the interval over which the convergence ap-

plies is finite. It is therefore of interest to inquire what are the corresponding

facts in a case where the interval is infinite. Such a case is afforded by the

Gram-Charlier series, which, in addition to having an infinite interval, is of

interest and importance as a mathematical statement of the law of errors ft

and because of its application to the representation of frequency functions.

In the following pages we propose to examine the degree of convergence

of the Gram-Charlier series and associated expansions, principally from the

point of view of developments in characteristic solutions of homogeneous

linear differential systems, a procedure which correlates the work closely

with the previously cited papers on degree of convergence. The results

obtained indicate that the Gram-Charlier series converges in general more

slowly than the Fourier series for a similar function in a finite interval.

Roughly speaking we may summarize the situation as follows: when the

remainder after n terms of a Fourier series is 0(1/«*) the remainder after n

terms of the Gram-Charlier series is 0(l/w*'2).

1. If we denote the normal probability function by faix) and its deriva-

tive of order n by <pnix), then

(1) fa(x) = i2Tt)-^e-Si*, faix) = H nix) faix),

* Presented to the Society, March 30 and June 21, 1929; received by the editors March 7, 1929.

t Jackson, these Transactions, vol. 13 (1912), pp. 491-515.

% Jackson, these Transactions, vol. 13 (1912), pp. 305-318.

§ Jackson, these Transactions, vol. 15 (1914), pp. 439-466.

|| Gronwall, these Transactions, vol. 15 (1914), pp. 1-30.

If Milne, these Transactions, vol. 19 (1918), pp. 143-156.

** Jackson, these Transactions, vol. 14 (1913), pp. 343-364, and vol. 22 (1921), pp. 158-166.See also Bulletin of the American Mathematical Society, vol. 27 (1921), pp. 415-431.

ft Charlier, Arkiv för Matematik, Astronomi, och Fysik, vol. 2 (1905).

422

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THE GRAM-CHARLIER SERIES 423

where Hn(x) is a polynomial of degree n, called an Hermite polynomial.*

The functions <pm(x) sind Hn(x) are biorthogonal in the infinite interval, so

that

(2) f <bm(x)Hn(x)dx = {J-K b

0 if m t¿ n,

.»! if m = n.

Hence we have the formal expansion of an arbitrary function <f>(x)

(3) <b(x) = a0tbo(x) + ai<j>i(x) + a2<j>i(x) + • • •

in which the coefficients are determined by the formulas

(4) an = (l/»0 J <b(x)Hn(x)dx.

The series (3) is the Gram-Charlier series. The convergence of this and

related series has been investigated by Myller-Lebedeff,| Watson, J Cramer,§

Szegö,|| Rotach,^[ Hille,** Stone,ft and others.

The functions <pn(x) and Hn(x) respectively satisfy the adjoint differential

equations

(5) u" + xu' + (X + 1)« = 0,

and

(6) v" - xv' + \v = 0,

for integral values of X, X =». If ^ is any solution of (6) the function

u = e~xi,2v

is a solution (5) and conversely. The transformations

u = e~x ¡iw, v = ex llw

carry (5) and (6) respectively into the self-adjoint equation tí

* Hermite, Comptes Rendues, vol. 58, p. 93.

f Myller-Lebedefi, Mathematische Annalen, vol. 64 (1907), p. 388.

X Watson, Proceedings of the London Mathematical Society, (2), vol. 8 (1910), pp. 393-421.

§ Cramer, Den Sjette Skandinaviske Matematikerkongres i Kjábenhavn, Kongresberetningen,

Copenhagen, 1926, pp. 399-425.|| Szegö, Mathematische Zeitschrift, vol. 25 (1926), pp. 112-115.

If Rotach, Promotionsarbeit, Zurich, Ceuf, 1925, 33 pp.

** Hille, Annals of Mathematics, (2), vol. 27 (1926), pp. 427-164.tt Stone, Annals of Mathematics, (2), vol. 29 (1928), pp. 1-11.

JI Equation (7) is sometimes called Weber's equation, and is satisfied by the harmonic functions

associated with the parabolic cylinder. See Weber, Mathematische Annalen, vol. 1, p. 1; Watson,

loc. cit.; Whittaker, Proceedings of the London Mathematical Society, vol. 35 (1902), p. 417.


424 W. E. MILNE [July

(7) w" + (X + 1/2 - x2/4)w = 0.

Corresponding to the three equations (5), (6), and (7) there are three distinct

expansions in terms of characteristic functions, each one of which can be

transformed into any other by means of the above relationships connecting

u, v, and w.

2. Instead of dealing directly with the series (3) it is convenient to use as

the basis of the following investigation the set of normalized orthogonal

solutions of (7),

Wo(x),Wi(x),W2(x),

defined by the equation

(8) Wn(x) = (27r)-1/4(«!)-1/2e-l2/4iin(x).

From the well known properties of Hermite polynomials we can easily write

down a number of important formulas as follows :

(9) (n + iy2Wn+i(x) + xWn(x) + nl'2Wn-i(x) = 0 ;

(10) W¿(x) = - (x/2)Wn(x) - n"2Wî(x) ;

(11) W¿(x) = (1/2)[(» + iy>2Wn+i(x) - n"2Wn-i(x)} ;

(12) W¿'(x) + (n + 1/2 - x2/4)Wn(x) = 0 ;

n-l

(13) ¿^Wm(x)Wm(s) = n^2{Wî(x)Wn(s) - Wn(x)Wî(s)}(x - s)~l ;m-=0

/(l ifWm(x)Wn(x)dx = |o

m — n,

m 9^ n.

From equation (11),

Wn+i(x) = (n + l)-1/2{2Wn'(x) + »«W_i(*)}.

If we form a similar equation for Wn-i(x) and substitute its value into the

above expression for Wn+i(x), then substitute again for Wn-3(x), and con-

tinue the process the following two formulas are obtained (in which we have

changed subscripts from «+1 to n), the first for the case in which n is even

and the second for the case in which n is odd :

(15) Wn(x) = 2[»-1'Wn'_1(x) + {(» - l)/(n(n - 2))} «W,'_,(*) + ■ • •

+ {(» - 1) • • • 7-5-3/(« • • • ó-^)}1'1»^*)]

+ {(«- 1) • ■ • 7-5-3-l/(»- • •6-4-2)}1/2e-*2/4(27r)-1/" ;

(16) Wn(x) = 2[»-1/W„'_i(x) + {(« - l)/(»(» - 2))}1'W„'_3(x) + • • •

+ {(»-!)•■ • 6-4-2/(w • ■ 5-34)j>'V,'W].


1929] THE GRAM-CHARLIER SERIES 425

Since the W's with odd subscripts vanish at 0 and °° we find from (15)

Wn(x)dx = (ir/2)l'4{ 1-3-5 •••(»- l)/(2-4-6 • ■ • n)}1'2,o

when « is even. On the other hand when n is odd (16) may be written

Wn(x) = 2{2-4-6-- •(«- l)/(l-3-5- • •»))",[H'i(*) + (l/2)1'W,,(i)

(18) +(l-3/(2-4)y'2Wl(x) + ---

+ {1-3-5 •••(«- 2)/(2-4-6 •••(«- l))]^2WLi(x)].

When m is even, Wm(x) vanishes at °o, but at x = 0 we have

(19) Wm(0) = (- 1)""2(2jt)-1/4(1-3-5 • • • (m - l)/(2-4-6 • • • m))1'2,

so that

(20) f Wn(x)dx-2(25r)-1'4[2-4-6 •••(»- l)/(l-3-5 • • • »)]1'2

X [1 - 1/2 + 1-3/(2-4) - l-3-5/(2-4-6) + • ■ •

± 1-3-5 •• • (n- 2)/(2-4-6- • • (n - 1))]

when « is odd. The series in brackets is the series, up to terms of degree

n — 1, for the expansion of (l+#2)-1'2 at the value x = l, and therefore the

sum of the terms in brackets in (20) differs from 2-1/2 by a quantity less than

the last term. Stirling's formula for w! enables us to estimate the magnitude

of this last term, which we find to be (2/(irw))1/2(l+0(«-1)). Therefore (20)

may be rewritten

(21) f Wn(x)dx = - (2/x)1'4[2-4-6 •••(»- l)/(l-3-5 • • ■ n)]l'\l + e),J o

when n is odd.

Finally if (17) and (21) are evaluated by means of Stirling's formula we

have

(22) I Wn(x)dx = ± «-1/4(1 + en), + if n is even, - if n is odd.Jo

3. To make clear the subsequent discussion it is needful to describe in

some detail the character of the function Wn(x). First of all Wn(x) is even

when n is even and odd when n is odd, so that we may limit the discussion

to the case where x is positive. By well known theorems of oscillation we

see from the differential equation (7) that in the interval 0 ^ x < h (where for

brevity we let A = (4w+2)1/2) the function Wn(x) oscillates with increasing

amplitudes, increasing intervals between the roots, and with decreasing slopes



at the roots. In the interval h<x<<x>, Wn(x) does not oscillate, but ap-

proaches zero to an infinite order as x becomes infinite. The roots of W„(x)

are identical with the roots of Hn(x), which are known to be all real, n in

number, symmetrically placed with respect to the origin, and all in the

interval —h<x<h. The largest maximum of Wn(x) occurs in the interval

r <x<h, where r denotes the largest root. The approximate location of this

largest root may be estimated as follows. In (7) we make the substitution

x = h - (2/hyiH,

which reduces it to

(23) d2w/dt2 + t[l - (2/hy>H/4]w = 0.

We now form two comparison equations

d2u/dt2 + tu = 0,

anddh/dt* + (t/2)v = 0.

We solve (23) with those conditions at / = 0 which furnish the solution

Wn(x) and solve the two comparison equations with the same initial values.

We denote the first positive roots of w, u, and v by /2, h, and /3 respectively.

Then if n is large enough that

1 - (2/hy'H3/4 > 1/2

we see by comparing the differential equations that

h<t2< t3.

Now h and t3 are not entirely independent of « because of the initial condi-

tions. But at / = 0, m and du/dt have opposite signs, from which we may

conclude that h has a lower bound C\, not zero, and independent of «. By

numerical calculation we find from the equation d2u/dt2+tu = 0 that it is

possible to take Ci = 71'3 = 1.91. Also t3 has a finite upper bound c3, independent

of n. Consequently

(24) h - (2/h)l'3c3 <r < h- (2/A)1«Ci.

To the function Wn(x) may be applied certain inequalities based on Hille's

investigations* in the case of Hermitian polynomials. Following his methods

we are able to show that

(25) | Wn(x) I < K(h2 - x2)-1«, -r£x£r,

where A is a constant independent of n. We have also

(26) | Wn(x) | < Kin-"12

* Hille, loc. cit., pp. 433-436.



for all values of x. A further inequality applicable to the interval h<x<<x>

may be established by the following considerations: When n is even, the func-

tion Wnix) is positive and decreasing and its curve is concave upward for

all values of x greater than h. If we denote by w0, wly w2, ■ ■ ■ , w¿, w{,

w2 , ■ ■ ■ the values of Wnix) and Wñ ix) at the points h, h+l, h+2, • • • ,

the following inequalities are apparent :

(27) w*_i > - w¿ .

From the equation (7) we see that, in the interval from h+k to h-\-k + l,

w" > \[ih + ky - h*]Wlc+1.

After integrating from h+k to h+k+1 and dropping some terms to

strengthen the inequality we have

(28) - wk' > %khwk+x.

From (27) and (28) it follows that

wk+i < 2wk-i/ikh)

and from this we get by successive substitutions

w2 < 2w0/h, Wi < 4w0/(3A2), w6 < 8w0/i3-5h3), • • • .

Therefore we may conclude that for x^h+2k

(29) | Wnix) I < k2n-"l\

The case in which n is odd is similarly treated.

4. In addition to estimates of the magnitude of W„ix) we shall require

estimates of the magnitude of the integral of Wnix). For this purpose let

fi, r2, r3, ■ • • , r denote the positive roots of Wnix). We then have the ine-

quality

(30) f Wnix)dx \<\ f ' l\V„ix)dx .

LetMi = ir/iri+i — n).

Then from (12) we have

(A2 - r^i)1'2 < 2tm < (A2 - r,3)1'2,

as may be seen by comparing the intervals between successive roots of those

solutions of the differential equations



d2u/dx2 + i(h2 - r2+1)u = 0,

dhi/dx* + \(h2 - x2)u = 0,

d2u/dx2 + \(h2 - n2)u - 0,

which vanish at x = r¿.

Hence there is a value x¿ of x between r¿ and r,-+i such that

(h2 - x?)"2 = 2tm.

The equation (12) may now be written in the form

W^(x) +m?Wn(x) = \(x2 - x?)Wn(x),

from which we obtain the integral equation

Wn(x) = (l/mi)Wn'(rn.i) sin tm(x - n+i)

(31)

4mi o r>+1

The integral of this from r¡ to ri+i is

fT*Wn{x)dx = - (I/WOW.'(r«n)

H-J dx f {sinw,(x - s)}(s2 - x*)Wn(s)ds.4niiJri Jri+i

Now

| s2 - x? I < r*+1 - r,i < 2irri+i/mi < 4irn+i(h2 - r/n)-1'",

and from (25)| ÎF.(i) | < A(Â2 - r^+i)-1'4

so that using these inequalities together with the inequalities for wz¡ we have

+ -— f {sin mix - s) \ (s2 - x? ) Wn(s)ds.4miJr...

< C2r*i(h2 - r?+i)-»\

I 1 Cr<+1 Cx

(32) - I dx I {smmi(x-s))(s2-x?)Wn(s)ds\4ntiJTi Jri+1

where C2 is a constant independent of n.

It will be found upon reference to Hille's arguments from which (25)

was derived that we also have the inequality

(33) | W7(*)| < K'(h2- x2y>\

where A' is independent of n, and x is between — r and r. Then using (33)

in connection with (32) we are led to the inequality

I W»(i(34) j Wn(x)dx < 8K'(h2 - r*+l)-*l* + C2ri+i(h2 - r^i)-»'«.



From the fact that the successive integrals satisfy the inequalities (30)

and alternate in sign we see at once that

(35) f W„(x)dxI Jo

< 8K'(h2 - r*)-3« + C2ri(h2 - r*)-°U, x ^ r,.

In particular if x is in the interval —N^x^N, where N = (3n+2)112,

(36) I fv n

Wn(x)dx < 8K'n-3ii + 0(n-,>i*).

Since the extrema of the integral occur only at the roots of Wn(x) its largest

maximum absolute value must be found either at one of the roots or else at

x = oo. If the former is true we find from (35) together with the fact that the

largest root is limited by (24) that

(37) \ Wn(x)dx < 2K'n-1«[l + «],J o

for all values of x. On the other hand if the largest maximum occurs at x = °o

its value is given by (22).

5. We are now in a position to investigate questions concerning degree

of convergence. First of all it will be interesting and instructive to consider

a series formed for a finite interval and to watch what happens as the ends of

the interval recede to — °o and °o. Together with equation (7) let us con-

sider a pair of Sturmian boundary conditions at the ends of a finite interval

(—c, c), and let w„(x) denote the normalized orthogonal characteristic solu-

tion of this system corresponding to the («+l)th characteristic number X„.

The formal expansion of an arbitrary function f(x) will be

(38) f(x) = c0w0(x) + CiWi (x) + c2w2 (x) + c3w3 (*) + ••••

This is a Sturm-Liouville series, so that the results obtained by Jackson re-

garding the degree of convergence of these series may be applied directly. We

are told for example that if f(x) has a continuous ¿th derivative of bounded

variation in the interval (—c, c), and if f(x) and its first k — 1 derivatives

vanish at — c and c, then the first n terms of the series (38) represents f(x)

with an error which is 0(Xn~*/2).

When c is allowed to become infinite the characteristic number \B+i ap-

proaches » as a limit and each term of the series (38) approaches* the corre-

sponding term of the series

* Cf. Weyl, Göttinger Nachrichten, 1910, p. 442; also Milne, these Transactions, vol. 30 (1928),

pp. 797-802.



(39) f(x) = CoWo(x) + CiWi(x) + C2W2(x) + C3W3(x) + ■■■

where

(40) Cn= ( f(x)Wn(x)dx.

We are thus led to conjecture that if f(x) has a &th derivative of bounded

variation in the infinite interval, and if f(x) and its derivatives vanish at

± oo, then the first n terms of (39) will represent f(x) with an error which is

0(n~kl2). The essential difference between this result and that for the

finite interval lies in the distribution of the characteristic numbers in the two

cases. For the finite interval the wth characteristic number is of the order

of magnitude of (irn/c)2, while for the infinite interval X„+i = n. Thus in the

former case the remainder term is 0(n~~k) while in the latter we conjecture

thatitisO(»-*/2).

Subsequent theorems show that this conjecture is substantially correct.

6. Instead of attempting to carry out rigorously the line of thought sug-

gested in the preceding paragraph we shall deal directly with the series (39).

Letting Sn(x) denote the sum of the first n terms of this series we may state

the first theorem as follows :

Theorem I. If f(x) has a continuous kth derivative of bounded variation,

and if xfik)(x), a;2/(*_1)(x), • • • , xk+lf(x) also are of bounded variation in the

infinite interval, then

Sn(x) = f(x) + 0(n~k'2), - N ^ x = N,

and

Sn(x) = f(x) + 0(»-<"-1)/6), - oo < x < oo .

These relations hold uniformly with respect to x in the specified intervals.

From (8) and (40) we may set

e-*2/4/(z)ff„(*)ciz,-«

whence integrating by parts k times with the aid of the relation

Hn(x)dx = - (n + I)"1 dHn+i(x),

and observing that the integrated terms vanish at the limits, we obtain

f* oo

C„ = [(n + 1)(» + 2) • • • (n + k)]-1'2 J Gk(x)Wn+k(x)dx,



in which

Gk(x) - e*'*-— [*-"«/(»)].dx"

From the hypotheses it is clear first of all that xGk(x) is of bounded varia-

tion in the infinite interval, and since f-k)(x) is also of bounded variation it is

easy to show that Gk(x) is of bounded variation in the infinite interval. Hence

we may set

Gk(x) = gi(x) - g2(x)

where gi(x) and g2(x) are positive or zero, continuous, monotone increasing,

and bounded. By the second law of the mean

XN »N nNGk(x)Wn+k(x)dx = gi(N) Wn+k(x)dx - g2(N) I Wn+k(x)dx

-N J (, J {

where £1 and £2 are in the interval ( — A, A). Therefore by (36)

)Wn+k(x)dx = 0(«-3/4)

Next we may write

J* 00 /* 00

Gk(x)Wn+k(x)dx = i [^^(^JflFn+^xVxJifx.AT J N

Since xG*(x) is of bounded variation, and since

f Gk(x)]J -N

f [WW*)/x]rfx = - (I/O f ÍF»+t(x)¿x

/» oo /» a

+ I x-2dx I lF„+i(x)¿x = 0(»-3/4)J j Jo

if £>A, we may deduce in the same manner as above that

J-\ ooGk(x)Wn+k(x)dx = 0(n~3ii).

N

The same is true of the integral from — oo to — N, so that finally

f Gk(x)Wn+k(x)dx = 0(W-"4)

andC = 0(«-fc/2-3'4).

Therefore in the interval ( — A, A) the inequality (25) gives



CnWnix) =0(«-*'2-1),

and hence in the same interval

00

T,CmWmix) =0(»-*'2).m=n

In view of the fact that under the given hypotheses the series is known to

converge to fix) the first conclusion stated by the theorem follows. To

establish the second we have only to use the inequality (26) in place of (25).

In the statement and proof of the foregoing theorem continuity of the /feth

derivative has been assumed. An examination of the methods of proof will

reveal that this restriction can be to some extent dispensed with. For

example the results can still be shown to hold true even iff(k)ix) has a finite

number of finite discontinuities.

It may also be observed that the other conditions of the hypotheses of

the theorem can be variously stated. Any set of conditions that will insure

the convergence of the series to fix) and at the same time make the functions

Gkix) and xGkix) of limited variation in the infinite interval will do.

From Theorem I we may derive directly a corresponding theorem ap-

plicable to the Gram-Charlier series (3).

Theorem II. If fax) has a continuous kth derivative of bounded variation,

and ifxex'i*fakKx), *V,'ty(*-1)(*), • • • , xk+le*'i'<t>ix)

are of bounded variation in the infinite interval, then

I 2n(*) - fax) | < il/»-*'2(l + *2)1/6<rl2/4,

where 2n(«) is the sum of n terms of (3) and M is a constant independent of n.

II fix) =extlifax) then fix) satisfies the hypotheses of Theorem I, and if

we multiply (39) by e~x'14 we obtain (3). From Theorem I we see that the re-

mainder after » terms of (3) will be

Oin-v^e-*'"

in the interval —N^x^N and will be

0(w-*/2+l/6)g-*2/4

outside this interval. Now when | x \ > N it is clear that

w1'6 < (1 + s2)1/6

whence the truth of Theorem II is apparent.

Since the function (l+#2)1/6e_I /4 is bounded we have



Corollary I. With the hypotheses of Theorem II

£„(x) = <b(x) + 0(n-k>2), - oo < x < oo .

It will be noted that Theorem I requires x*+l/(x) to be of bounded varia-

tion in the infinite interval, which carries with it the implication that for x

large

f(x) =0(x-*-1).

This condition can be somewhat lightened, at the expense however of a cor-

responding Increase in the remainder term for large values of x. We suppose

that/(x) satisfies the hypotheses of Theorem I, let

f(x) =Sn(x) + Rn(x),

multiply this by (1 +x2) and get rid of the x2 in the terms of the series by

two successive applications of the formula

xWp(x) = -(/> + íyiWp+lx) - p"2Wp-i(x).

Then we have

/i(x) = C¿Wo(x) + ClWi(x) + ■■■ + C„'_2PF„_2(x) + Ri.n(x),

in whichfi(x) = (l + x2)/(x),

C; = [(p - DpY'Vp-, + (2p + 2)CP + [(p + l)(p + 2)Yi2Cp+2,

and^i.n(x) ={[(«- 2)(« - l)]1'2^ + 2nCn-i}W„.i(x)

+ {[(»- i)nY"Cn-t + (2n + 2)Cn}Wn(x)

+ {^(n+Dyi^n-ljWnMx)

+ {[(»+ l)(n + 2)]U2Cn}Wn+i(x)

+ F„(x)(l + x2).

In the course of the proof of Theorem I we have shown that

Cn = o(«-*/2-3'4)

and this fact together with (25), (26), and the inequality

w1'6 < (1 + x2)1'6, |*|>A,

enables us to show that

Fi.„(x) = 0(n~k'2)(i + x2)7'6.

Use of the recursion formula shows that the coefficients which we have

designated by Cp' are actually the coefficients which we would have obtained



in the formal expansion of the function/i(x) itself, and therefore Ai,„(x) is

the remainder after n terms of the series (39) formed for the function /i(x).

Therefore we have the following modification of Theorem I.

Theorem III. If f(x) has a continuous kth derivative of bounded variation,

and if xk~lf(x), xk~2f'(x), x*-3/"(x), • • • , xrlfw(x) are also of bounded

variation when x is large, then

Sn(x) = /(*) + 0(«-*'2)(l + x2)7'6,

for all values of x.

It is this theorem which is to be regarded as closest analogue to the

theorem of Jackson on Sturm-Liouville series referred to above.

The statement and proof of the corresponding theorem for the case of

the series (3) may be left to the reader.

7. Since the results obtained in the preceding paragraph indicate a rate

of convergence much slower than in the corresponding situation for Fourier

series the question naturally arises whether these conclusions are inevitable

or are merely due to inadequate methods of proof. The following example

shows that no substantial improvement in the conclusions stated by the

theorems need be expected.

Let/(x) = | x |c~x,/4. This function satisfies the hypotheses of Theorem I

for k = 1 except that the derivative has a finite discontinuity at x = 0. But

we have seen that the proof of Theorem I can still be carried through in spite

of a finite number of discontinuities of the derivative. We multiply the

identity (13) by the function | 5 \e~'2'*, integrate with respect to 5 from — oo to

oo, assume that is odd, then set x = 0, and have

5.(0) = »"'MO) [/ ~ / 1 e"'l4Wn(s)ds.

From equations (1) and (8)

»1/2 [ / - J" 1 f'i*W»(s)ds = 2Wn-i(0)

so that

5.(0) = 2Wn2-i(0) = 2(25r)-1/2[l-3-5 •••(«- 2)/(2-4-6 •••(«- 1))].

The use of Stirling's formula shows that this last expression is equal to

(2/7r)»-1/2 + 0(«-3'2).

Since/(0)=0I 5.(0) -/(0) | > (1/tt)«-1'2.



This example therefore shows that a stronger conclusion than the one stated

for Theorem I is not to be expected.

As a second example designed to show that the hypotheses about vanish-

ing at infinity cannot be dispensed with, consider the function f(x) = 1, which

satisfies all the conditions of Theorem I for any k except that of vanishing

at infinity. We note that C„ = 0 if n is odd, and C„ is twice the expression

given by (17) if n is even. Therefore if we integrate (18) from — oo to x and

compare coefficients with the above values of C„ we find that

x)dx,

or

Sn(x) = (7r/2)1'4[l-3-5 • • -n/(2-4-6- • (n - I))]1'2 f V„(

Sn(x) = «^"(l+O f Wn(x)dx,J -n

where n is odd. From this it is plain that

lim Sn(x) = 0,

so that for any given n we may always choose x so large that

f(x) - Sn(x) > 1 - €, e > 0.

Thus no relation of the form

| f(x) - Sn(x) | = 0(n-"2)

can hold uniformly in the infinite interval.

We may remark however that by means of (36) and (20) it may be shown

that

(41) S„(x) = »"4(1 + «„) F f Wn(x)dx + f Wn(x)dx~\

= 1 + 0(n-^2) + nl'*(l + €„) f Wn(x)dxJo

= 1 +0(«-1/2), - N ^ x ¿ N,

= 1 +0(1), - oo < x < co.

8. Theorems I, II, and III have stated sufficient conditions for a given

degree of convergence. The next two will furnish certain necessary condi-

tions. The first of these two theorems is closely analogous to one given by

Jackson* for the case of Fourier series and shows that the existence of deriva-

* Jackson, Über die Genauigkeit der Annäherung stetiger Funktionen durch rationale Funktionen

gegebenen Grades und trigonometrische Summen gegebener Ordnung, Dissertation, Göttingen, 1911;

Satz XVIII.


436 W. E, MILNE [July

tives of certain orders is necessary for a given degree of convergence.

Theorem IV. If a series

S(x) = AWM + AiWi(x) + A2W2(x) + ■■■

converges so that for any n its remainder after n terms is

Rn(x) + 0(n~k~'), k an integer, e > 0,

uniformly for x in any interval, then the sum S(x) has a continuous derivative

of order 2k in the same interval.

Consider the sum

n+v n+p

YjnqAmWm(x) = 2>«[£„,(*) - Rn+^x)].

By partial summation this may be put in the form

n+jH-1

2 [(« + 1)« - m<]Rm+i(x) + n"Rn(x) - (n + pyRn+p+i(x).mmmtt

Now

(m + 1)« — mq < q(m + l)«-1.

Hence the general term under the sign of summation is less in absolute value

thanMq(m + l)«-i-*-«, M a constant,

from which we see that the sum itself is

0(m«-*-«).

The two terms not under the summation sign are clearly also

0(n*-k-')

so that

"'S, m*AmWm(x) =0(n«-x-').

From this we conclude that the set of series

00

2>MmWm(x) (q = 1,2, • • • , k)m—0

all converge uniformly in the given interval.



Now from equation (12)

oo oo

¿2AJV¿'(x) = - T,Am(m + 1/2 - x2/4)Wm(x)

00 GO

= - T,mAmWm(x) + (x2/4 - 1/2) ¿ZAmWm(x).m—0 m=0

From the uniform convergence of the two series on the right follows the

uniform convergence of the series of second derivatives, which in turn

establishes the existence and continuity of the second derivative of S(x).

To proceed, we differentiate twice the relation

S"(x) = (x2/4 - h)S(x) - ÍnAnWn(x),n—0

substitute for W„" (x) from (12), and obtain formally

S<4>(x) = (x2/4 - §)S"(x) + xS'(x) + $S(x)

00 00

+ d - x2/4) ¿ZnAnWn(x) + ¿Zn2AnWn(x).n—0 n—0

The uniform convergence of the series involved in this last relation shows

that the differentiation was legitimate and establishes the existence and

continuity of 5(4)(x). In the same manner successive differentiations and

substitutions establish the existence of the derivatives of even order up to

Sl2k)(x). It is only necessary to eliminate by means of preceding equations

all of the series which appear with coefficients involving x, since otherwise

the double differentiation would introduce terms containing Wn'(x), which

cannot be expressed in terms of Wn(x).

Corollary. In a fixed finite interval Theorem IV may be applied to the

series (3) without change of wording.

Since we may take our interval as large as we like, it appears that the

corollary establishes the existence of derivatives up to order 2k for all values

of x provided the hypotheses hold uniformly in the infinite interval.

Our second theorem furnishing necessary conditions deals with the be-

havior of the function at infinity, and is as follows :

Theorem V. 7/ the series

S(x) = A0Wo(x) + AiWi(x) + A2W2(x) + A3W3(x) + ■■■

converges so that the remainder after n terms is 0(n~hli) uniformly in the infinite

interval, thenS(x) =0(\x\~k)

when | x | ¿5 large.



It may be verified without difficulty that the functions Wnix), Wn'-iix),

Wn-2ix) all have the same sign when x>h, so that from the relationship

n^Wnix) -in- íyi'Wn-tix) = 2W¿-iix),

obtained from (11), we derive the inequality

| Wn-2ÍX) | < «1/2(W - 1)-1/2| Wnix) \ .

By successive repetitions this gives us

I Wn-tmix) | < [nin - 2) ■ ■ ■ in - 2m + 2)Y"

■ [in - l)in - 3) •••(»- 2m + l)]-,/2| IP.(*) \ < «1/2| Wnix)\.

I llrp'lXJ'ICA

| Wn-2m+lix) | < W1'2 | Wn-lix) | •

Now if we let X = h+2k+6 and choose | x | = X we may apply inequality

(29), with the exponent —k/l — i/2, to the functions Wn-iix) and Wnix),

so that in view of the inequalities just obtained above

(42) Wmix) =0(«-i/2-1), m = 0,1,2, •••, n, \x\^X,

and from this followsSnix) =0(rc-*/2).

By hypothesisSix) = Snix) + Oin-*i*),

so thatSix) = 0(«-*'2) = OiX~k).

Since this result is true in particular when | x \ = X, we easily draw the con-

clusion of the theorem. The fact that X takes on isolated values only does

not affect the truth of the theorem for the continuous variable x.

9. Up to the present we have been concerned with the degree of con-

vergence of the expansions (3) or (39) in which the coefficients are determined

by the formulas (4) or (40). Now, however, we take up the case in which the

function/(#) is to be expressed as a linear combination of W0, Wi, ■ ■ ■ , Wn,

with coefficients which are not necessarily given by (40). The first result is

Theorem VI. If fix) satisfies a Lipschilz condition

|/(*i) -fix2)\ <L\xi- xt\and if

I */(*) I < L,then there exists a sum

(43) 2„ix) = AonWoix) + AmWiix) + • • • + AnnWnix)



such that| fix) - S.(x) | < BLn-"2,

in which B denotes a constant independent of n and off(x).

We shall prove the theorem only for even values of n, n = 2m, though it

will be clear that the result holds without this restriction. Our starting point

is a theorem due to Jackson* which asserts the existence of a polynomial

F\m(x) of degree m at most such that in a closed interval of length /

|/(x) - Fim(x)| < BiLJmr1.

Here as in subsequent formulas Bi, B2, etc. denote constants independent of

« and of f(x). We shall choose the interval

-Xájál, X = h + 8,so that

/ = 2[(8m + 2)1'2 + 8] = i32m)li*[l + Oinr1'*)].

We may therefore write

(44) Plm(x) =/(x) + pi»(x),

where| Pimdx) | < B2Lm-V2.

We next turn to the expansion of unity in a series 5m(x) as given by (41).

Clearly we have

Sm(x) = e-*2/4P2m(x),

where P2m(x) is a polynomial of degree m at most, so that (41) can be put in

the form

(45) e-*2'4P2m(x) = 1 + P2m(x)

with the conditions

| Pim(x) | < B3m-"2, - N á x á N,

andI Pimix) | < Bi, - » < X < OO .

Now we multiply (44) and (45), obtaining

e-^4F.(x) =f(x) + R„ix),

in which P„(x) is a polynomial of degree n at most. From the inequalities

satisfied by pi„(x) and p2.(x) together with the fact that |x/(x) | <L we

readily deduce that

* These Transactions, vol. 14 (1913), p. 351, Theorem V.


440 W. E. MILNE [Juyl

(46) \Rn(x)\ < BLn-1'2.

Since e~xt/iPn(x) is a linear combination of Wo, Wi, ■ ■ ■ , Wn, we may set

e~*2'*Pn(x) = Z„(x)

and have

(47) 2B(x) = /(x) + Rn(x)

in the interval — X^x^X, so that the desired theorem is established in this

interval.

Now we consider the magnitude of the remainder term for x outside of

this interval. First of all we observe by means of (42) for the case k = 0, that

Xx ( 1 + 0(iWi(x)Wj(x)dx = {

.x \0(n~2),

l+0(n-2), i=j£n,

i < j ^ n.

If A in denotes the largest coefficient in (43) we multiply (47) by W¡(x), inte-

grate from — X to X, and have

Ajn(l+0(n-i)) = f f(x)Wj(x)dx+ f Wj^RMdx,J-x J-x

from which it appears that A ,„ is bounded with respect to n. For from (26)

and (29) it is seen that \xlleWj(x) \ is bounded, and the same is true of

I*/(*)'!> s0 that the first integral is bounded. That the second integral is

bounded is obvious from (46).

Finally we use (42) again for the case k = 1 getting

| Wp(x) | < 55»-3'2 t>-0,l, • • , »).

whence

| 2„(x) | < BeLn-"2,

and since \f(x) \ <Ln~112 when | x | >X, we see that, for a suitable B, (46)

holds for all values of x, and the theorem is established for the infinite interval.

If x is replaced by 21/2a; a sum of the type (43) is converted into a sum of

the type

(48) ct„(x) = Ôo<po(z) + ôi<Pi(x) H-h bn<t>n(x),

whence we have

Theorem VIL Theorem VI also holds for the sum (48).

On the other hand the introduction of the factor e~x',4: enables us to deduce

from Theorem VI the following



Theorem VIII. If<b(x) satisfies a condition of the form

| ez?l4<b(xi) - e^'^xù \ <L\xi- x2\

and if| xe^'Wx) | < L

there exists a sum of the type (48) such that

| <b(x) - an(x) | < BLn-Me-**'4.

10. We are now in a position to establish a theorem very similar to one

proved by Jackson* for the indefinite integral of a trigonometric sum.

Theorem IX. Let<j>(x) be an integrable function for which

(49) f <b(x)dx = 0,t/ -co

and suppose that there exists a sum a„(x) of the form given by (48) such that

(50) <b(x) = On(x) + Pn(x),

where| Pn(x) | < 6e-2'4

for all values of x in the infinite interval. Then

(51) I <b(x)dx = co<t>a(x) + ci<f>i(x) + • • • + cn<bn(x) + rn(x)J-CC

in which| rn(x) | < SBtn-^e-^i*.

The proof is so similar to that of Jackson that it might be omitted, were

it not that the behavior of the remainder terms when x is large requires some

slight additional investigation. We shall sketch the proof briefly. If we

integrate (50) from — oo to oo and use (49) we find that the term bo<bo can be

included with pn(x) to form a new remainder p„' (x) for which

|p»'(*)| < (21/2 + l)e6-*2'4.

We then integrate (50) from — oo to x, obtaining

(52) J <b(x)dx = bi<bo(x) + b2<bi(x) + • • • + bn<bn-i(x) + Rn(x)

n(x) = I Pn(x)dx.

in which

1 These Transactions, vol. 13 (1912), pp. 491-515.



Clearly A„(x) vanishes at — oo and in view of (49) it also vanishes at oo.

Therefore

/CO

p„'(x)¿x.

If x is negative we use the first form of A.(x), but if x is positive we use the

second form. Suppose that x is positive. Then

e-^lWx

/ooe-*2'4xdx

= 2(2»" + Oear1«-**'4 < Six-1*-*'*.

The same is true when x is negative. Thus we have shown that A„(x) satisfies

the second part of the hypothesis of Theorem VIII with L = 5e. That A„(x)

satisfies the first part of the hypothesis with L = 5e follows from the fact, read-

ily shown from the foregoing inequalities, that the derivative of e*2/4An(x)

does not exceed 5e in absolute value.

Therefore from Theorem VIII we see that there exists a sum sn(x) involv-

ing the first n <p's, such that

| Rn(x) - snix) | < SBm-UH-Sn.

We use this result in connection with (52), combine the two sums and

obtain (51).

From the two theorems VIII and IX we derive

Theorem X. 7/</>(x) has a ik — 1) th derivative which satisfies the conditions

| exIs/Y(*-i)(a.1) _ e^ityk-i)^ \<l\xi- x2\ ,

(53) \xexl'W<-»ix)\ <L,

and if

(54) lim <p(x) = 0,x=± «

then there exists a sum <r.(x) of the form (48) such that

| 0(x) - ffn(x) | < iSByLn-xih-Ji*.

To establish this we first apply Theorem VIII to the function <p<-k-l)ix),

then, use Theorem IX to obtain an approximating sum for <£(*_2)(x), again

use IX to obtain an approximating sum for 0(*_3,(x), and continue thus till



the desired result is reached for fax). In making these applications of

Theorem IX we need to know, in addition to the stated hypotheses, that

fa^ix)dx = 0 im - 1,2, ••-,*- 1),-00

or what amounts to the same thing, that

lim fam>ix) =0 im = 0,1,2, • ■ • , k - 2).X=± oo

Now if x is large and positive

0(t-i> =C - f fak-»ix)dx

= C + 0(*-2e-*2'2),

0<*-3> - C* + C' + Oix-3e-x*'2).

Continuing thus we find that fax) may be expressed as a polynomial with

an added term which is Oix-ke~x2'2). But the condition (54) shows that the

polynomial must vanish identically, and therefore C, C', etc. all vanish, which

completes the proof.

If in Theorem X we make the change of function

fax) =e-x*"fix),

we are enabled to state a theorem regarding approximation by sums of the

type (41):

Theorem XI. If fix) has a ik — l)th derivative which satisfies a Lipschitz

condition

| Fk~»ixi) - /<*-»(*,) | < L | xi - ¡e» | ,

and if| ̂ mf(.m>tx) \<L (m = 0,1,2, • ■ • , k - 1),

then there exists a sum 2„(x) such that

Snix) = fix) + 0(«-*'2)

uniformly in the infinite interval.

The proof of this theorem may be left to the reader.

Stanford University,

Stanford University, Calif.


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